Similarity

Similar figures

In Euclidean geometry, two objects are similar if they have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotation and reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other.

Translation

Rotation

Reflection

Scaling

For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other.

Figures shown in the same color are similar If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure.

Two congruent shapes are similar, with a scale factor of 1. However, some school textbooks specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different if the triangles are to qualify as similar.

Similar triangles

Two triangles, △ABC and △A′B′C′, are similar if and only if corresponding angles have the same measure: this implies that they are similar if and only if the lengths of corresponding sides are proportional. It can be shown that two triangles having congruent angles (equiangular triangles) are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem. Note that the "AAA" is a mnemonic: each one of the three A's refers to an "angle". Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent.

There are several statements each of which is necessary and sufficient for two triangles to be similar:

• The triangles have two congruent angles, which in Euclidean geometry implies that all their angles are congruent. That is:

If ∠BAC is equal in measure to ∠B′A′C′, and ∠ABC is equal in measure to ∠A′B′C′, then this implies that ∠ACB is equal in measure to ∠A′C′B′ and the triangles are similar.

• All the corresponding sides have lengths in the same ratio:

${\tfrac {AB}{A'B'}}={\tfrac {BC}{B'C'}}={\tfrac {AC}{A'C'}}$. This is equivalent to saying that one triangle (or its mirror image) is an enlargement of the other.

• Two sides have lengths in the same ratio, and the angles included between these sides have the same measure. For instance:

${\tfrac {AB}{A'B'}}={\tfrac {BC}{B'C'}}$ and ∠ABC is equal in measure to ∠A′B′C′.

This is known as the SAS similarity criterion. The "SAS" is a mnemonic: each one of the two S's refers to a "side"; the A refers to an "angle" between the two sides.

When two triangles △ABC and △A′B′C′ are similar, one writes

△ABC ∼ △A′B′C′.

There are several elementary results concerning similar triangles in Euclidean geometry:

• Any two equilateral triangles are similar.
• Two triangles, both similar to a third triangle, are similar to each other (transitivity of similarity of triangles).
• Corresponding altitudes of similar triangles have the same ratio as the corresponding sides.
• Two right triangles are similar if the hypotenuse and one other side have lengths in the same ratio. There are several equivalent conditions in this case, such as the right triangles having an acute angle of the same measure, or having the lengths of the legs (sides) being in the same proportion.

Given a triangle △ABC and a line segment ${\overline {DE}}$ one can, with ruler and compass, find a point F such that △ABC ∼ △DEF. The statement that the point F satisfying this condition exists is Wallis's postulate and is logically equivalent to Euclid's parallel postulate. In hyperbolic geometry (where Wallis's postulate is false) similar triangles are congruent.

In the axiomatic treatment of Euclidean geometry given by G.D. Birkhoff (see Birkhoff's axioms) the SAS similarity criterion given above was used to replace both Euclid's Parallel Postulate and the SAS axiom which enabled the dramatic shortening of Hilbert's axioms.

Similar triangles provide the basis for many synthetic (without the use of coordinates) proofs in Euclidean geometry. Among the elementary results that can be proved this way are: the angle bisector theorem, the geometric mean theorem, Ceva's theorem, Menelaus's theorem and the Pythagorean theorem. Similar triangles also provide the foundations for right triangle trigonometry.

Other similar polygons

The concept of similarity extends to polygons with more than three sides. Given any two similar polygons, corresponding sides taken in the same sequence (even if clockwise for one polygon and counterclockwise for the other) are proportional and corresponding angles taken in the same sequence are equal in measure. However, proportionality of corresponding sides is not by itself sufficient to prove similarity for polygons beyond triangles (otherwise, for example, all rhombi would be similar). Likewise, equality of all angles in sequence is not sufficient to guarantee similarity (otherwise all rectangles would be similar). A sufficient condition for similarity of polygons is that corresponding sides and diagonals are proportional.

For given n, all regular n-gons are similar.

Ratios of sides, of areas, and of volumes

The ratio between the areas of similar figures is equal to the square of the ratio of corresponding lengths of those figures (for example, when the side of a square or the radius of a circle is multiplied by three, its area is multiplied by nine — i.e. by three squared). The altitudes of similar triangles are in the same ratio as corresponding sides. If a triangle has a side of length b and an altitude drawn to that side of length h then a similar triangle with corresponding side of length kb will have an altitude drawn to that side of length kh. The area of the first triangle is, $A={\tfrac {1}{2}}bh$, while the area of the similar triangle will be $A'={\tfrac {1}{2}}(kb)(kh)=k^{2}A$. Similar figures which can be decomposed into similar triangles will have areas related in the same way. The relationship holds for figures that are not rectifiable as well.

The ratio between the volumes of similar figures is equal to the cube of the ratio of corresponding lengths of those figures (for example, when the edge of a cube or the radius of a sphere is multiplied by three, its volume is multiplied by 27 — i.e. by three cubed).

Galileo's square–cube law concerns similar solids. If the ratio of similitude (ratio of corresponding sides) between the solids is k, then the ratio of surface areas of the solids will be k², while the ratio of volumes will be k³.

Licensing

Content obtained and/or adapted from:

• Similarity (geometry), Wikipedia under a CC BY-SA license